Compactness Properties Preserved under Projection Mapping
Theorem
Let $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$ be a sequence of topological spaces.
Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \langle {\left({X_\alpha, \vartheta_\alpha}\right)}\right \rangle$.
Let $\operatorname{pr}_\alpha: \left({X, \vartheta}\right) \to \left({X_\alpha, \vartheta_\alpha}\right)$ be the projection on the $\alpha$ coordinate.
Then $\operatorname{pr}_\alpha$ preserves the following compactness properties.
That is, if $\left({X, \vartheta}\right)$ has one of the following properties, then each of $\left({X_\alpha, \vartheta_\alpha}\right)$ has the same property:
- Compactness
- Sigma-Compactness
- Countable Compactness
- Sequential Compactness
- Lindelöf Space
- Local Compactness
- Paracompactness
Proof
First note that Projection from Product Topology is Continuous.
Also note that Projections are Surjections.
It follows from Compactness Properties Preserved under Continuous Surjections that:
are all preserved under projections.
Next note that Projection from Product Topology is Open.
It follows from Local Compactness Preserved under Open Continuous Surjections that local compactness is preserved under projections.
The final result is that Paracompactness Preserved under Projections.
Finish these results off.
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 3$: Invariance Properties
